f(x) and f′(x) are continuous, differentiable functions that satisfy
f(x)=x^3+4x^2+∫(from 0 to x)(x−t)f′(t) dt.
What is f′(5)−f(5)?
To find the value of f'(5) - f(5), we'll need to first differentiate both sides of the given equation and then substitute x = 5 into the resulting expressions for f'(x) and f(x).
Differentiating both sides of the equation:
d/dx [f(x)] = d/dx [x^3 + 4x^2 + ∫(0 to x) (x - t) * f'(t) dt]
Using the Fundamental Theorem of Calculus, we can differentiate the integral term:
d/dx [∫(0 to x) (x - t) * f'(t) dt] = (x - x) * f'(x) + ∫(0 to x) (1 * f'(t)) dt
Simplifying the above equation:
f'(x) = x^2 + 1 * ∫(0 to x) f'(t) dt
Substituting x = 5 into the equation:
f'(5) = 5^2 + 1 * ∫(0 to 5) f'(t) dt
Now, let's find the value of the integral term:
∫(0 to 5) f'(t) dt
Since f'(t) is continuous and differentiable, we can use the Fundamental Theorem of Calculus again to evaluate the integral:
∫(0 to 5) f'(t) dt = F(5) - F(0)
Here, F(x) represents an antiderivative of f'(x).
Since we don't have an explicit expression for f'(x), we can't determine the exact value of this integral without further information.
Therefore, we can't calculate the value of f'(5) - f(5) without knowing the specific function f(x) or having more information.